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I would like to know if the Lebesgue spaces $L_p$ with $ 0 < p < 1 $ are separable or not.

I know that this is true for $1 \leq p < + \infty$, but I do not find any references for the

case $ 0 < p < 1 $.

Thank you

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2 Answers

As Chandru1 stated tentatively, the same arguments apply for $0< p<1$ as for $1\leq p<\infty$, so it is no surprise that a separate proof is hard to find. For example, an introduction to $L^p$ spaces on subsets of $\mathbb{R}^n$ can be found in Chapter 8 of Measure and integral by Wheeden and Zygmund. A proof of separability is outlined for $1\leq p<\infty$ in Theorem 8.15. Part of Theorem 8.16 asserts separability for $0< p<1$, but for proof they simply refer to the proof of 8.15.

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Do you happen to know where $L^p$ space for $0 < p < 1$ are used? Or are the solely of theoretical interest? I mean the ones where we don't use the counting measure, I do see a use for the $\ell^p$, $0 < p < 1$ (it's on wikipedia). –  Jonas Teuwen Aug 31 '10 at 7:04
    
@Jonas: No I don't, but you may be interested in the following if you haven't seen it: mathoverflow.net/questions/28147/… –  Jonas Meyer Aug 31 '10 at 15:37
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Please refer this article. It talks about $L_{p}$ spaces for $0 < p \leq 1$. Link: http://www.jstor.org/stable/2041603?seq=2

Look at the step functions, the ones that take rational values and whose steps have rational endpoints there should be only countably many of those. And then you can perhaps apply the same argument, you use to prove it for $L_{p}$ spaces for $1 < p < \infty$.

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Can you please provide some key ideas from the link next time you answer a question like this? Links can die, or the user may not be able to access the subscription, or the OP may not find which part of the article is really relevant. See meta.stackexchange.com/questions/7515/why-is-linking-bad/… for detail. –  KennyTM Aug 16 '10 at 11:18
    
@Kenny TM: Oh, correct! –  anonymous Aug 16 '10 at 11:27
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